Trifylfots

Here’s a simple fractal created by dividing an equilateral triangle into smaller equilateral triangles, then discarding (and rotating) some of those sub-triangles, then doing the same to the sub-triangles:

Fractangle (triangle-fractal) (stage 1)


Fractangle #2


Fractangle #3


Fractangle #4


Fractangle #5


Fractangle #6


Fractangle #7


Fractangle #8


Fractangle #9


Fractangle (animated)


I’ve used the same fractangle to create this shape, which is variously known as a swastika (from Sanskrit svasti, “good luck, well-being”), a gammadion (four Greek Γs arranged in a circle) or a fylfot (from the shape being used to “fill the foot” of a stained glass window in Christian churches):

Trifylfot


Because it’s a fylfot created ultimately from a triangle, I’m calling it a trifylfot (TRIFF-ill-fot). Here’s how you make it:

Trifylfot (stage 1)


Trifylfot #2


Trifylfot #3


Trifylfot #4


Trifylfot #5


Trifylfot #6


Trifylfot #7


Trifylfot #8


Trifylfot #9


Trifylfot (animated)


And here are more trifylfots created from various forms of fractangle:













































Elsewhere other-accessible

Fractangular Frolics — more on fractals from triangles

Hour Re-Re-Re-Re-Powered

In “Dissing the Diamond” I looked at some of the fractals I found by selecting lines from a dissected diamond. Here’s another of those fractals:

Fractal from dissected diamond


It’s a distorted and incomplete version of the hourglass fractal:

Hourglass fractal


Here’s how to create the distorted form of the hourglass fractal:

Distorted hourglass from dissected diamond (stage 1)


Distorted hourglass #2


Distorted hourglass #3


Distorted hourglass #4


Distorted hourglass #5


Distorted hourglass #6


Distorted hourglass #7


Distorted hourglass #8


Distorted hourglass #9


Distorted hourglass #10


Distorted hourglass (animated)


When I de-distorted and doubled the dissected-diamond method, I got this:

Hourglass fractal #1


Hourglass fractal #2


Hourglass fractal #3


Hourglass fractal #4


Hourglass fractal #5


Hourglass fractal #6


Hourglass fractal #7


Hourglass fractal #8


Hourglass fractal #9


Hourglass fractal #10


Hourglass fractal (animated)


Elsewhere other-engageable:

 

Hour Power
Hour Re-Powered
Hour Re-Re-Powered
Hour Re-Re-Re-Powered

Dissing the Diamond

In “Fractangular Frolics” I looked at how you could create fractals by choosing lines from a dissected equilateral or isosceles right triangle. Now I want to look at fractals created from the lines of a dissected diamond, as here:

Lines in a dissected diamond


Let’s start by creating one of the most famous fractals of all, the Sierpiński triangle:

Sierpiński triangle stage 1


Sierpiński triangle #2


Sierpiński triangle #3


Sierpiński triangle #4


Sierpiński triangle #5


Sierpiński triangle #6


Sierpiński triangle #7


Sierpiński triangle #8


Sierpiński triangle #9


Sierpiński triangle #10


Sierpiński triangle (animated)


However, you can get an infinite number of Sierpiński triangles with three lines from the diamond:

Sierpińfinity #1


Sierpińfinity #2


Sierpińfinity #3


Sierpińfinity #4


Sierpińfinity #5


Sierpińfinity #6


Sierpińfinity #7


Sierpińfinity #8


Sierpińfinity #9


Sierpińfinity #10


Sierpińfinity (animated)


Here are some more fractals created from three lines of the dissected diamond (sometimes the fractals are rotated to looked better):



















And in these fractals one or more of the lines are flipped to create the next stage of the fractal:




Previously pre-posted:

Fractangular Frolics — fractals created in a similar way

Dissecting the Diamond — fractals from another kind of diamond

Circus Trix

Here’s a trix, or triangle divided into six smaller triangles:

Trix, or triangle divided into six smaller triangles


Now each sub-triangle becomes a trix in its turn:

Trix stage #2


And again:

Trix #3


Trix #4


Trix #5


Trix divisions (animated)


Now try dividing the trix and discarding sub-triangles, then repeating the process. A fractal appears:

Trix fractal #1


Trix fractal #2


Trix fractal #3


Trix fractal #4


Trix fractal #5


Trix fractal #6


Trix fractal #7


Trix fractal (animated)


But what happens if you delay the discarding, first dividing the trix completely into sub-triangles, then dividing completely again? You get a more attractive and symmetrical fractal, like this:

Trix fractal (delayed discard)


And it’s easy to convert the triangle into a circle, creating a fractal like this:

Delayed-discard trix fractal converted into circle


Delayed-discard trix fractal to circular fractal (animated)


Now a trix fractal that looks like a hawk-god:

Trix hawk-god #1


Trix hawk-god #2


Trix hawk-god #3


Trix hawk-god #4


Trix hawk-god #5


Trix hawk-god #6


Trix hawk-god #7


Trix hawk-god (animated)


Trix hawk-god converted to circle


Trix hawk-god to circle (animated)


If you delay the discard, you get this:

Trix hawk-god circle (delayed discard)


And here are more delayed-discard trix fractals:







Various circular trix-fractals (animated)


Post-Performative Post-Scriptum

In Latin, circus means “ring, circle” — the English word “circle” is actually from the Latin diminutive circulus, meaning “little circle”.

Fractangular Frolics

Here’s an interesting shape that looks like a distorted and dissected capital S:

A distorted and dissected capital S


If you look at it more closely, you can see that it’s a fractal, a shape that contains itself over and over on smaller and smaller scales. First of all, it can be divided completely into three copies of itself (each corresponding to a line of the fractangle seed, as shown below):

The shape contains three smaller versions of itself


The blue sub-fractal is slightly larger than the other two (1.154700538379251…x larger, to be more exact, or √(4/3)x to be exactly exact). And because each sub-fractal can be divided into three sub-sub-fractals, the shape contains smaller and smaller copies of itself:

Five more sub-fractals


But how do you create the shape? You start by selecting three lines from this divided equilateral triangle:

A divided equilateral triangle


These are the three lines you need to create the shape:

Fractangle seed (the three lines correspond to the three sub-fractals seen above)


Now replace each line with a half-sized set of the same three lines:

Fractangle stage #2


And do that again:

Fractangle stage #3


And again:

Fractangle stage #4


And carry on doing it as you create what I call a fractangle, i.e. a fractal derived from a triangle:

Fractangle stage #5


Fractangle stage #6


Fractangle stage #7


Fractangle stage #8


Fractangle stage #9


Fractangle stage #10


Fractangle stage #11


Here’s an animation of the process:

Creating the fractangle (animated)


And here are more fractangles created in a similar way from three lines of the divided equilateral triangle:

Fractangle #2


Fractangle #2 (anim)

(open in new window if distorted)


Fractangle #2 (seed)


Fractangle #3


Fractangle #3 (anim)


Fractangle #3 (seed)


Fractangle #4


Fractangle #4 (anim)


Fractangle #4 (seed)


You can also use a right triangle to create fractangles:

Divided right triangle for fractangles


Here are some fractangles created from three lines chosen of the divided right triangle:

Fractangle #5


Fractangle #5 (anim)


Fractangle #5 (seed)


Fractangle #6


Fractangle #6 (anim)


Fractangle #6 (seed)


Fractangle #7


Fractangle #7 (anim)


Fractangle #7 (seed)


Fractangle #8


Fractangle #8 (anim)


Fractangle #8 (seed)


At the Mountings of Mathness

Mounting n. a backing or setting on which a photograph, work of art, gem, etc. is set for display. — Oxford English Dictionary

Viewer’s advisory: If you are sensitive to flashing or flickering images, you should be careful when you look at the last couple of animated gifs below.


H.P. Lovecraft in some Mountings of Mathness






Agnathous Analysis

In Mandibular Metamorphosis, I looked at two distinct fractals and how you could turn one into the other in one smooth sweep. The Sierpiński triangle was one of the fractals:

Sierpiński triangle


The T-square fractal was the other:

T-square fractal (or part thereof)


And here they are turning into each other:

Sierpiński ↔ T-square (anim)
(Open in new window if distorted)


But what exactly is going on? To answer that, you need to see how the two fractals are created. Here are the stages for one way of constructing the Sierpiński triangle:

Sierpiński triangle #1


Sierpiński triangle #2


Sierpiński triangle #3


Sierpiński triangle #4


Sierpiński triangle #5


Sierpiński triangle #6


Sierpiński triangle #7


Sierpiński triangle #8


Sierpiński triangle #9


When you take away all the construction lines, you’re left with a simple Sierpiński triangle:


Constructing a Sierpiński triangle (anim)


Now here’s the construction of a T-square fractal:

T-square fractal #1


T-square fractal #2


T-square fractal #3


T-square fractal #4


T-square fractal #5


T-square fractal #6


T-square fractal #7


T-square fractal #8


T-square fractal #9


Take away the construction lines and you’re left with a simple T-square fractal:

T-square fractal


Constructing a T-square fractal (anim)


And now it’s easy to see how one turns into the other:

Sierpiński → T-square #1


Sierpiński → T-square #2


Sierpiński → T-square #3


Sierpiński → T-square #4


Sierpiński → T-square #5


Sierpiński → T-square #6


Sierpiński → T-square #7


Sierpiński → T-square #8


Sierpiński → T-square #9


Sierpiński → T-square #10


Sierpiński → T-square #11


Sierpiński → T-square #12


Sierpiński → T-square #13


Sierpiński ↔ T-square (anim)
(Open in new window if distorted)


Post-Performative Post-Scriptum

Mandibular Metamorphosis also looked at a third fractal, the mandibles or jaws fractal. Because I haven’t included the jaws fractal in this analysis, the analysis is therefore agnathous, from Ancient Greek ἀ-, a-, “without”, + γνάθ-, gnath-, “jaw”.

Mandibular Metamorphosis

Here’s the famous Sierpiński triangle:

Sierpiński triangle


And here’s the less famous T-square fractal:

T-square fractal (or part of it, at least)


How do you get from one to the other? Very easily, as it happens:

From Sierpiński triangle to T-square (and back again) (animated)
(Open in new window if distorted)


Now, here are the Sierpiński triangle, the T-square fractal and what I call the mandibles or jaws fractal:

Sierpiński triangle


T-square fractal


Mandibles / Jaws fractal


How do you cycle between them? Again, very easily:

From Sierpiński triangle to T-square to Mandibles (and back again) (animated)
(Open in new window if distorted)


Allus Pour, Horic

*As a rotating animated gif (optimized at ezGIF).


Performativizing Paronomasticity

The title of this incendiary intervention, which is perhaps my most contrived title yet, is a paronomasia on Shakespeare’s “Alas, poor Yorick!” (Hamlet, Act 5, scene 1). “Allus” is a northern form of “always”, “pour” has its standard meaning, and “Horic” is from the Greek ὡρῐκός, hōrikos, which strictly speaking means “in one’s prime, blooming”. However, it could also be interpreted as meaning “hourly”. So the paronomasia means “Always pour, O Hourly One!” (i.e. hourglass).

A Seed Indeed

Like plants, fractals grow from seeds. But plants start with a small seed that gets bigger. Fractals start with a big seed that gets smaller. For example, perhaps the most famous fractal of all is the Koch snowflake. The seed of the Koch snowflake is step #2 here:

Stages of the Koch snowflake (from Fractals and the coast of Great Britain)


To create the Koch snowflake, you replace each straight line in the initial triangle with the seed:

Creating the Koch snowflake (from Wikipedia)


Animated Koch snowflake (from Wikipedia)


Now here’s another seed for another fractal:

Fractal stage #1


The seed is like a capital “I”, consisting of a line of length l sitting between two lines of length l/2 at right angles. The rule this time is: Replace the center of the longer line and the two shorter lines with ½-sized versions of the seed:

Fractal stage #2


Try and guess what the final fractal looks like when this rule is applied again and again:

Fractal stage #3


Fractal stage #4


Fractal stage #5


Fractal stage #6


Fractal stage #7


Fractal stage #8


Fractal stage #9


Fractal stage #10


I call this fractal the hourglass. And there are a lot of ways to create it. Here’s an animated version of the way shown in this post:

Hourglass fractal (animated)